"so I'll try putting some distortion pedals in the path while I'm there and see what difference they make" - GordonC
Gordon, any distortion should certainly change the quality of the output - but some will be far more effective than others, and the effect will depend on the pitch of what you are playing and how the TM is set up.. I suppose what I am warning is that, to gain any usable insight, you may be in the shop for quite a while - and if your memory is anything like mine, take written notes!
A lot of distortion is based on clipping, with varying degrees of 'softness' or 'hardness' largely defined by sharpness (slew rate) of the distorter.. These tend to give varying increase primarily in odd harmonics.. the article below shows a useful map of harmonic positions for vowel formants - This mapping IME is quite 'rigid' and the formant positions do not seem to change much, if at all, with the pitch one excites the filter with.
I think this is why I have found non-symetrical pulse waveforms best on my experiments - and these are not a common distortion even for grunge! I played with a circuit quite similar to the Enkelaar distortion, and this gave me the best results - So I would cart your enkelaar to the shop if I was you.. However, I had the advantage of being able to adjust the 2 formants wherever I wanted - I have no idea how the TM operates WRT formant adjustment - it might even have algorythms which generate 'required' harmonics from the input signal ( listening to Peters performance I am inclined to think it must ) in which case everything I have said is probably wrong!
http://www.geofex.com/article_folders/sing-wah/sing-wah.htm
Fred.